Algebraic and transcendental factors

As another example where integration by parts is useful (and, in fact, necessary), consider the integral

Choosing fails, as in the previous (counter)example, since the resulting integral is more difficult than the original. Instead:

Let and .

Thus and .

Then

In this case, the second term in the final expression requires another application of integration by parts:

Let and .

Thus and .

Then

Substituting the last expression into the previous result:

Note that if the second integration by parts step had instead used and , this would have "undone" the first step and we would have ended up with an integrand very much like the one we started with:

Thus, the correct choice of and is particularly important when multiple applications of the technique are required. In general, if is chosen to be an algebraic function in the first step, it should be algebraic in all subsequent steps.

Choosing and

Fortunately, there is a mnemonic for choosing and , which covers a large variety of integrands:

This mnemonic only works when the integrand is the product of two different types of factors. The factor whose type of function appears higher in this list should generally be chosen as , the factor whose type appears lower as .

For example, in the integral

the choices should be , since this is a logarithmic function, and , since this is an algebraic one ("L" appears before "A" in the mnemonic).
On the other hand, in the integral

If the mnemonic doesn't seem to work for a given integral it is possible that it may be a simpler form that can be evaluated using the substitution method, or perhaps rewritten into a simpler form using algebraic or trigonometric techniques (e.g., trigonometric identities).

A slightly different mnemonic that works almost as well — and has the added benefit of sounding more like an English word — is:

← L I P E T →

Here the "P" stands for Power, which includes polynomials and roots (fractional powers). The other letters are as above.

Notice that the last two letters are switched in this form; this is usually not an issue, since integrals involving a product of trigonometric and exponential factors can generally be done "either way" (with respect to the choice of and ) or not at all using this technique.